Posted: Feb 16, 2012 2:21 pm
The Hanging Monkey wrote:CarlPierce wrote:You use the small change idea to find the gradient at a particular point.
And the gradient at any point will give you the general rule?CarlPierce wrote:The idea being that the smaller and smaller the change is the better and better the answer is
You mean as dx tends towards zero, then ignoring the dx^2 term becomes a better approximation, giving a more accurate answer?
I think I have it
My maths is by no means bad, but it is very piecemeal. I haven't studied maths formally since I was 16 but I've picked up bits here and there throughout my degree and Ph.D. It's very satisfying to have some of the many gaps filled in.
Thats right you do 'have it'. In general as dx is very small, mathematically we say as it tends to zero - you can ignore all but lowest power of it.
so another example consider the curve y = x^3
for a small change dx in x, y increases by = (x + dx)^3 - x^3 = x^3 + 3x^2.dx + 3x.dx^2 + dx^3 - x^3 = 3x^2.dx + 3x.dx^2 + dx^3
as dx -> 0 we can disgard the dx^2 and dx^3 terms so dy tends towards = 3x^2.dx
so dy/dx = 3x^2